Problem 123 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Which of the following functions have period \(2 ?\) (A) \(\\{x\\}+\cos \pi x\) (B) \(\tan \left(\frac{\pi}{2}[x]\right)\) (C) \(\sin x+\\{x\\}\) (D) \(\sin (\cos x)\)

Step-by-Step Solution

Answer

None of the functions have a period of 2.

Step 1: Understanding the Period Definition

The period of a function is the smallest positive value \( T \) such that \( f(x + T) = f(x) \) for all \( x \). We are given multiple functions and need to find if any of them have a period of 2.

Step 2: Analyzing Function A

Function A is \( \{x\} + \cos(\pi x) \). The sawtooth function \( \{x\} \) has period 1 because it repeats every integer, and \( \cos(\pi x) \) has a period of 2, as \( \cos(\pi (x+2)) = \cos(\pi x) \). Since \{x\} does not have a period of 2, the combination does not either.

Step 3: Analyzing Function B

Function B is \( \tan \left( \frac{\pi}{2} [x] \right) \). The function \( [x] \) is the floor function, and \( \tan \left( \frac{\pi}{2} n \right) \) is undefined for odd integers and periodic with period 2 for integers. Therefore, function B does not have a well-defined period of 2.

Step 4: Analyzing Function C

Function C is \( \sin x + \{x\} \). The \( \sin x \) function has a period of \( 2\pi \) and \{x\} has a period of 1. Neither of these match a period of 2, nor does their combination result in a period of 2.

Step 5: Analyzing Function D

Function D is \( \sin(\cos x) \). The \( \cos x \) function has a period of \( 2\pi \), and the \( \sin \) of any value is also periodic, but the inside function \( \cos x \) ensures the function \( \sin(\cos x) \) does not achieve a period of 2.

Step 6: Conclusion

None of the presented functions have a period of 2 upon analysis. They either have no well-defined period or periods that do not match 2.

Key Concepts

Sawtooth Function

The sawtooth function, often represented as \( \{x\} \), is a fundamental piecewise linear periodic function. Its key characteristic is that it produces a linear sawtooth wave, rising from zero to one, and suddenly dropping back to zero. The function repeats itself for every integer

• The period of the sawtooth function is 1. This means it repeats every unit interval along the x-axis.
• Graphically, it looks like successive teeth of a saw, hence the name.
• In mathematical terms, it's defined as \( \{x\} = x - \lfloor x \rfloor \), where \( \lfloor x \rfloor \) is the floor function.

Understanding the behavior of \( \{x\} \) is crucial because it affects combinations with other functions like sine, cosine, or even more complex forms. Even though it's useful, this function does not possess a period of 2, as requested in the original exercise.

Floor Function

The floor function, denoted as \( [x] \) or \( \lfloor x \rfloor \), plays an essential role in understanding periodicity within functions.

• This function takes a real number and returns the greatest integer less than or equal to that number, effectively 'rounding down' to the nearest whole number.
• For example, \( \lfloor 3.7 \rfloor = 3 \) and \( \lfloor -2.5 \rfloor = -3 \).
• Because it focuses on integer values, periodic behavior involving the floor function often ties into how other functions react over integer intervals.

While helpful in building integer-based function frameworks, the floor function alone or in certain combinations does not exhibit a definitive period of 2. The intricacies of its integer-stepping nature sometimes complicate function periodicity when paired with trigonometric elements.

Sine Function

The sine function, \( \sin x \), is one of the most renowned periodic functions in mathematics, known for its smooth wave-like oscillations.

• The fundamental period of \( \sin x \) is \( 2\pi \).
• This means that the wave repeats its values every \( 2\pi \) units along the x-axis, producing regular peaks and troughs at specific intervals.
• The amplitude (or height) of the wave is consistently 1, and it oscillates between -1 and 1.

When combined with other functions like the sawtooth function, understanding each component's period is crucial to determining whether the overall function maintains or changes the periodicity. It was analyzed whether the combination of \( \sin x \) with \( \{x\} \) could result in a period of 2, but these functions combined do not achieve that requirement.

Cosine Function

Like its sine counterpart, the cosine function, \( \cos x \), shares a similar rhythmic, periodic nature notable in mathematical and physical contexts.

• The cosine function also has a period of \( 2\pi \), meaning it repeats itself every \( 2\pi \) units.
• Graphically, \( \cos x \) starts at its maximum value (1), decreases to -1, and then returns to 1 over each period.
• This characteristic waveform leads to it closely resembling a shifted version of the sine wave.

This function in isolation or when multiplied inside expressions like \( \cos \pi x \) incorporates a period of 2 due to the factor \( \pi \). It was analyzed alongside functions like the sawtooth, but a mismatch of baseline periods meant the desired period of 2 couldn’t be met as a whole for combined functions in the exercise.

Tangent Function

The tangent function, represented as \( \tan x \), is a unique periodic function in trigonometry known for its distinct properties and behavior compared to sine and cosine.

• The standard period of the tangent function is \( \pi \), repeating its cycle every \( \pi \) units rather than \( 2\pi \).
• Tangent is undefined at odd multiples of \( \frac{\pi}{2} \); it sharply rises towards infinity or decreases to negative infinity at these points.
• Instead of peaks and troughs, tangent exhibits rising and falling slopes between its undefined points, making it unique among common trigonometric functions.

In context, when \( [x] \) values are introduced as components of \( \tan \left( \frac{\pi}{2} [x] \right) \), the function does not cleanly organize to display a period of 2. Its inherent characteristics often lead to discontinuities that can complicate periodicity analysis.